Skip to main content
#
Full text of "Cerenkov radiation from bunched electron beams"

■ id d ac y RESEARCH REPORTS DIVISION NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA 93943 bort Number NPS-61-83-003 IIBRARY f e IS DIVISION naval; : ate school MONTEREY, CALIFORNIA 9394Q NAVAL POSTGRADUATE SCHOOL Monterey, California CERENKOV RADIATION FROM BUNCHED ELECTRON BEAMS F. R. Buskirk and J.R. Neighbours October 19 82 Revised April 1983 Technical Report Approved for public releaser distribution unlimited repared for: FEDDOCS hief of Naval Research D 208.14/2:NPS-61 -83-003 rlington, Virginia 22217 NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral John Ekelund David Schrady Superintendent Provost The work reported herein was supported by the Chief of Naval Research. Reproduction of all or part of this report is authorized. This report was prepared by: UNCLASSIFIED 'ittUHlTY CL ASSlFlt ATIONOF Inlt> PtOE iWUm Dmlm tnlmtmj) RE PORT DOCUMENTATION PAGE 1 REPONT NUMBER NPS-61-83-003 GOVT ACCESSION NO * TITLE (mnd Subitum) Cerenkov Radiation from Bunched Electron Beams 7. AUTMO«C«J F. R. Buskirk and J. R. Neighbours » PERFORMING ORGANIZATION NAME AND AOORESS Naval Postgraduate School Monterey, CA 93940 II. CONTROLLING OFFICE NAMC AND AOORESS Chief of Naval Research Arlington, Virginia 22217 rr KEAD INS7 WUCTIONS UtHiKh i UMI'LETING KOkM J RECIPIENT'S CATALOG NUMBER 5 type of report * perioo coveaeo *, Technical Report • PERFORMING ORG. REPORT NUMftA . _J I CONTRACT OR GRANT NUMBERS 10 PROGRAM ELEMENT. PROJECT, TASK AREA * iOHK UNIT NUMtfNS 62763N;RF68-342-800 N0001482WR20195 12. REPORT DATS November 19 8 2 11 NUMBER OF PAGES MONlTOAlNO ACCNCV NAMC A AOORESSflf dlttmtmnt from Controlling Olllcm) <» SECURITY CLASS, (ol inn import) Unclassified It* DECLASSIFICATION' OOWNGRACTfiG - SCHEDULE It. DISTRIBUTION STATEMENT (ol «ii» Report; Approved for public release; distribution unlimited 17. DISTRIBUTION STATEMENT (ol thm •».rr»cl antmtmd In Slock 20. II dlllmrmnt Itom Rmport) 1*. SUPPLEMENTARY NOTES It- KEY WORDS (Contlnum on tmvmtmm mldm II n«C(li«ry «j Idmnllty oy block numbmr) Cerenkov Radiation Microwave Radiation Bunched Electrons 20. ABSTRACT (Cominum on tmvmtmm mldm It nmemmsmry mnd Idmnllty 6r mloe* m~a>o«<i Cerenkov radiation is calculated for electron beams which exceed the velocity of radiation in a non dispersive dielectric medium. The electron beam is assumed to be bunched as emitted from a travelling wave accelerator, and the emission region is assumed to be finite. Predictions include (a) emission at harmonics of the bunch rate, (b) coherence of radiation at low frequencies (c) smear- ing of the emission angle for finite emission regions, (d) explicit evaluation of cower sroectrum in terms of bunch dimensions. The do ,; FORM AN 7) 1473 EDITION OF I NOV «t IS OBSOLETE S/N 102-0 14- 660 1 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whmn Dmlm Sntmtmdi UNCLASSIFIED CfaCURITV CLASSIFICATION OF THIS PAGECWi^n Dmtm Entmrmd) results apply to microwave emission from fast electrons in air or other dielectrics. DD Form 1473 (BACK) S/N 1 "SfolV-SeO! UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(TWi«n Dmtm Entm TABLE OF CONTENTS Page Introduction 1 Calculation of Poynting Vector 3 Fourier Coefficients of the Current 6 Vector Potential 8 Radiated Fower 11 Cerenkov Angle 13 Discussion of Results 15 a. Effect of Pulse Size 15 b. Shearing of the Cerenkov Angle 16 c. Behavior at high Frequencies Related 16 to Pulse Parameters. Concluding Remarks; 18 References 20 APPENDICES A. Derivation of Cerenkov Radiation Al for a Single Pulse of Charge B. Derivation of Equation 7 Bl C. Temporal Structure of the Electron CI Pulse from a Travelling Wave Linear Accelerator D. Form Factors Dl Distribution List I NTRODUCTION The radiation produced by gamma rays incident on ordinary dielectric materials such as glass was first discovered by Cerenkov 1 in 1934 and was described in terms of a charged particle (electron) moving faster than light in the medium by Frank and Tamm 2 in 1937. A summary of work to 1958 is contained in the treatise by Jelly-*. An important application is the Cerenkov particle detector which is familiar in any particle physics laboratory, and an early and crucial application occurred in the discovery of the antiproton 4 - . Because the distribution of intensity of Cerenkov radiation is proportional to the frequency, the radiation at microwave frequencies would be low unless beams are intense and bunched so that coherent radiation by many electrons contributes . Danos^ in 1955 calculated radiation produced by a planar beam passing above a dielectric interface and a hollow cylindrical beam passing through a hole in a dielectric. Experimental and theoretical investigations at microwave frequencies were reviewed by Lashinsky^ in 1961. This investigation was motivated by a recent renewed interest which has included the study of stimulated Cerenkov radiation, in which the electron may be in a medium consisting of a gas' or a hollow dielectric resonator^* 9. Recent developments of electron accelerators for applications such as free electron lasers (FEL) have aimed toward high peak currents in bunches in contrast to nuclear and particle physics applications, where low peak but high average currents are desirable to avoid saturating 1 detectors. The high peak currents in the new accelerators should yield enhanced Cerenkov radiation, as is calculated in this paper . CALCULATION OF THE POYNTING VECTOR In the following derivation, we consider the Cerenkov radiation produced in a dispersionless medium such as cases or other dielectrics, by a series of pulses of electrons such as are produced by a traveling wave electron accelerator (Linac). The pulses or bunches are periodic, the total emission region is finite and the bunches have a finite size. In determining the radiated power, the procedure is to calculate the Poynting vector from fields which are in turn obtained from solutions of the wave equations for the potentials. Since the current and charge densities entering into the wave equations are expressed in fourier form the resulting fields and radiated power also have fourier components. In the derivation, r is the coordinate at. which the fields will be calculated, r i.s the coordinate of an element of the charge which produces the fields and n is a unit vector in the direction of r. We assume that E(r,t) and B(r,t,) have been expanded in a fourier series;, appropriate for the case where the source current 1.3 periodic. Then we have 00 E (tt) -£ e" iaJt 1 tf,a») oj=-°° (1) and a corresponding expansion for B, where w is a discrete _& — ^ variable and E and B are fourier series coefficients. The poynting vector S is given by S = -~E x~B (2) and it is easy to show that the average of S in a direction given A by a normal vector n i: T oo i r n.s at - i y. -lu. -J /i\=— oo oj ) x B(r,-(D) 0)=-» (3) where T is an integer multiple of the period of the periodic current . Letting c = (ye) "1/2 be the velocity of light, in the medium, the wave equations for A,4> and their solutions are, 2 3 -*■,-*> -*■ » V — 9 — \ A(r,t) = yj(r,t) c z 3t z i I c 2 dt 2 i '?ir,t) = 1/e p(r,t) m 3 , A(r,t) =y D(r-r', t-t ') J(f',t ') dr 'dt ' «f> (r,t) = 1 e D(r-?',t-t , )p(r' ,t')d 3 r'dt' (4) (5) where the Green's function D is given by D(r,t) = 1_ 5 (t-r/c) 4irr (6) The vector potential A(r, t) also can be developed in a fourier series expansion of a form similar to (1) with an expression for the fourier series coefficients given by T A(r,a>) = 1 r dtA(?,t)e 1 ' 1 f dtA(r,t) T I ■ill d 3 r> j(r' r ai)_l 1__ e iaJ l r - r ' ' /c r-r' (7) Now if we assume that the observer is far from the source so that jr j>> |r' | for regions where the integrand in (7) is important we can let |r-r | = *r - n • r in the exponential and jr - r' = : in the |r - r 1 |~1 factor in (7), obtaining (where n = r/r) A(r,o>) = Ji_ e^ r/c r[fa 3 r'j;(r' ,a>)e- iiai/crn - t ' 47rr JJJ (8) The fourier series coefficients of the fields are obtained from those for the vector potential (8) through the usual relations B = V x A and E = -V<|> - 9A . Under the conditions 3t leading to (3) the field fourier coefficients are-^: -* - ^ -l ^ B(r,uj) = ico n x A ( r , oj ) c (9) -* -i -A -* -k E(r,co) = -c n x B(r,oj) (10) The poynting vector can now be found by using (9) and (10) in expansions like (1) and then substituting in (2). However it is more convenient to deal with the frequency components of the radiated power by substituting (9) and (10) into the expression of the average radiated power (3). T 00 1 [ n-Sdt = 1 ) T J y ^— oj n x A ( r , oo ) c (11) F OURIER COMPONENTS OF THE CURRENT The expression (7) for the fourier components of the vector potential contains the fourier components of the current density. Consequently it is necessary to examine the form of the current and its fourier development. Assume the current is in the z direction and periodic. If the electrons move with velocity v, and we ignore for the moment the x and y variables, the charge or current functions should have the general form f(z,t) = £ e lk z Z £ e ia)t f(k z ,a») (12) (13) Z CO Under the condition of rigid motion, f (z,t) = f (z-vt) it is easy to show that ,(k ,cj) = 6 . f (k ) , % z co,kv~ z (14) where fvkj = e 1K z z f (z)dz *v o Z — z (15) Thus the restrictions of equation (13) reduce the two dimensional fourier series of eq. (12) to essentially a one dimensional series (14). With (14) in mind, the current density associated with the electron beam from a linear accelerator should be periodic in both z, t, with a fourier series expansion, but the x and y dependence should be represented by a fourier integral form: 4«o J_(r,t) ~ vp(r,t) = (2 TT) — CO dk dk x y f ,, \~ i (k-r - cat) J dk y ^_ e Po k =- c z (k) 16) where the fourier components of the charge density are P,o (k) = dx dy - | dze U 7 ) p (r) is p(r,t) evaluated at t = o and J is assumed to be in the z direction. Note in eq. (16) that k z and ^ are both discrete and from (14), oj = k z v. VECTOR POTENTIAL The results of the previous section can be applied to the evaluation of the vector potential and in turn to the fields. Let the infinite periodic pulse train be made finite, extending from z = -Z ' to z = +Z ' and let 9 be the angle between n and A. Then the cross product in (11) can be written I A *t #"*" \ I a V ioor/c n x A(r,oj) = smfl -7 — e CO GO 7 ' f dx' / dy' /" dz' e " ifi ^ ,a)/c Jm J>™ =^>7 I JSOO .= 00 — 2 k =- c t-At2 dk / dk ) v0 o (£)<$, e 1K ' r (2n) z I x J ot *y k ^ = _^ - k z v /W (18) But Z* j. * dx' J dy' / dz' e ir '- (k ™ /c) -Z' (2tt) 2 5(k x - n x aj/c) 5 (k - nu/c)I(Z') ( 19 ) where Z' T ,„ n / , , i(k - n co/c)z' •> I(Z') = / dz' e z z 2 sinGZ . (20) J-Z' G and G = k z - n 2 go /c = 00 /v - n z co /c And thus the cross product term is /\ •*■ ^ 1 Tj icor/c n x A(r,a>) I = sinG-^r e v£o(n x Vc, n oi/c, o)/v)I(Z') (21) Note that. o> is a discrete variable but from 19, the continuous variables k x and k y become evaluated at discrete points . Returning to (17), a more symmetric form may be obtained by assuming rhat p (r)# which is periodic in z with period Z, is actually zero between the pulses. Denoting by p ' ( r) the charge density of a single pulse, which is zero for z < o and z > Z the integral on z can be written z z , -ik z ,-». f , -ik z ',■»> -, -ik z ' ,a dz e z p (r; = dz e z p (r) = dz e z p (r) J (22) Then (17) the fourier coefficient-, of the charge density, becomes oo p (k) =| |fj r d 3 r e" ik ' r : '(r) = |pl ik) (23) where p n '(k) is the three dimensional fourier transform of the single pulse decribed by p Q ' (r) . Substituting these expressions into (21) gives a final simple result, for the cross product, form: |n x A(r,<o)l = sine H e lajr/C (v/Z) Pe ' (k)I(z') 47tr - (24) where I(Z' ) = | sin GZ' G G = oj/v - n co/c (25 ) k = (n oo/cn a)/c,oo/v) The components of the Cerenkov E and B fields may now be found by substituting (24) in (9) and (10). 10 RADIATED POWER The frequency components of the average radiated power are obtained by substituting (24) into (II). The negative: frequency terms equal the corresponding positive frequency terms , yielding a factor of 2 when the summation range is changed. Multiplying by r^ converts to average power per unit solid angle, dP/dft , yielding T 00 f§=r 2 i I" n-Sdt = r 2 l£V |nx A(?, M )| 2 00 =1 C W(<o,n) (26) where w(a)/n) is defined to be 2 w (a) ,n) = — 2 g- sin 9 (v /Z ) | p ' (kj j I (Z ) (4it) (27) W(o),n) is the power per unit solid angle radiated at the frequency 0), which is a harmonic of the basic angular frequency uj of the periodic pulse train. To find P , the total power radiated at. the frequency co , W is multiplied by dfi and integrated over solid angle. Note that n z = cos 9, and as 9 varies, G changes according to (25), 11 w ith dG = - (co/c) dn so that d^ = dc{> (c/oj) dG (28) Noting that the integral over <£ yields 2"^ , we find the result for the total radiated power at the frequency oj for all angles G" p = r_ _ 2 2 }J_ 60 v ^ 4tt c 2 J sin 2 6 jPo(k) | 2 I 2 (Z') 5 dG (29) G' 12 CERENKQV ANGLE The remaining integral over G may now be examined. The sin^ S and p factors may often be slowly varying compared to the l2(z') factor, the latter being shown in Fig. 1. For large Z', the peak in l2(z') becomes narrow, and if the integrand may be neglected outside the physical range G'<G<G", G I, I (Z ')dG = |4(Z ') / i, 2 sinGZ '} GZ' 'dG = 4tt (30) Then, evaluating the sin 9 factor and p Q (k) at the point corresponding to G = 0, (which is cos 6 = n z = c/v) shows that. 9 at. the peak of I(Z' ) is the usual Cerenkov angle 9 C . We thus obtain for large Z' P = £- covsin 6 |p (k)| 4ttZ'/Z oj 4lT C l ~° ' (31) Now let 2Z'/Z = ratio of the interaction length to pulse spacing N, the number of pulses. Also Z = v2 tt/ o^ or 2tt/z = '^ / v so that, (in the large Z' limit), 2 ** 2 P oj = 4tt" 000 °o vsin 9 c IP=^ k )i N. (32) To compare with usual formulations, (32) is divided by Nv to obtain the energy loss per unit path length per pulse: dE = y_ dx 4tt o)co sin 8 I p (k) , , 1 2 (33) 13 If the pulse is in fact, a point charge, the fourier transform Dq (k) reduces to q, the total charge per pulse and (33) is very similar to the usual Cerenkov energy loss formula, where for a single charge q. the radiation is continuous and the factor oo cj in (33) is replaced by u) d go. In the present case the pulse o train is periodic at angular frequency oj q and the radiation is emitted at the harmonic frequenies denoted by oj . 14 DISCUSSION OF RESULTS Equation (29) and the approximate evaluation expressed as (32) form the main results. Some consequences will now be noted. a. EFFECT OF PULSE SIZE. The spatial distribution of the charge in the pulse appears in p^lk) , which is the fourier transform of the charge distribution. The peak of I 2 (2') in figure 1 occurs at G = or n z = c/v. Thus at the peak, co/v = n z oj/c i\o that k, the argument of p' (k), is evaluated at (from 25) s+S "*■ A k = noo/c (34) We may also define a charge form factor F(k) o;<k) = q p(ki (3S) The form factor F(k) is identically one for a point charge, and for a finite distribution F(k) = 1 for k =o . Furthermore F(k) must fall off as a function of k near the origin if all the charge has the same sign. If the pulse were spherically symmetric, F(k) would behave as elastic electron scattering form factors defined for nuclear charge distributions 1-1 . In that case, the mean square radius <r 2 > of the charge distribution is given by the behavior of F(k) near the origin. F(k) ■*■ 1 - <r 2 > k 2 /6 (spherical pulse) (36) 15 b. SMEARING OF THE CERENKOV ANGLE. For a finite region over which emission is allowed, namely if 2Z ' is finite, the function I 2 (Z'), appearing in the integral in (29), will have a finite width. Since the peak height is 4Z ' 2 and the area is 4tts ' , (30), we can assign an effective width 2T = area/ height - ^ I z ' or r=7T/2z' (37) Thus the radiation is emitted mainly near G = o (which corresponds to 9 = 9 C ) hut in a range Ag = +T . But from (25), 00 , 00 . . . AC = — in 7 = — a(cos9) so that there is a range in cos0 over c ^ c wr. ich emission occurs: a / Q^ C 7T (38) A (cos9) = - ^—r 00 ZZj Note that the finite angular width of the Cerenkov cone angle in (38) has the factor l/co , indicating that the higher harmonics are emitted in a sharper cone. c. 3EHAVI0R AT HIGH FREQUENCIES RELATED TO PULSE PARAMETERS. To be specific let the charge distribution for a single pulse be given by gaussian functions 1 .-* 2 2 ? 2 9 p (r) = A exp(-x/a - y /a - z7b ) Then F(k) may be found 39) F(k) = exp(-k 2 a 2 /4 -k 2 a 2 /4 -k 2 b 2 /4) y Z (40) Beam pulse parameters could then be determined by measuring the Cerenkov radiation. For example, fast electrons from an accelerator in air will e ait with a 9 C of several degrees in which case k x and k y in (40) can be neglected, giving F(k) = exp(-k z 2 b 2 /4) = exp[-co 2 b 2 /(4v 2 ) ] (41) The expected behavio:: of P as a function of go is shown 00 qualitatively in Fig. 2 as a linear rise at. low frequencies followed by a fall off at. higher frequencies, the peak occurring at (o = v/b m (42) Furthermore, a different behavior would be expected at. very high frequencies. The formulation from the beginning represents coherent, radiation from all charges, not only from one pulse, but. coherence from pulse to pulse. F(k) then describes interference of radiation emitted from different, parts of the pulse, but note that expressions (29} and (32) will still be proportional to q2 - n 2 e 2 w here n ± s the number of electrons in a pulse. Thus the n^ dependence of P indicates coherence. But. above CO some high frequency go j_ such that oj . /c - 2\\ I % , where I is the mean spacing of electrons in the cloud, the radition should switch over to incoherent radiation from each charge and P should be 00 proportional to n. The incoherent radiation should then rise again as a function of au . 17 CONCLUDING REMARKS The general results presented here describe the Cerenkov radiation produced by fast electrons produced by a linear accelerator. For an S band Linac operating at about 3Ghz (10 cm radiation) , the electron bunches are separated by 10 cm and would be about 1 cm long at 1% energy resolution. Microwave Cerenkov radiation is expected and has been seen in measurements at the Naval Postgraduate School Linac. Two types of measurements were made. In measurements of Series A, an X-band antenna mounted near the beam path, oriented to intercept the Cerenkov cone, was connected to a spectrum analyser. Harmonics 3 through 7 of the 2.85 GHz bunch frequency were seen but power levels could not be measured quantitatively. Harmonics 1 and 2 were below the wave guide cut off. In the series B measurements, the electron beam emerged from the end window of the accelerator, and passed through a flat metal sheet 90 cm downstream oriented at an angle <j> from the normal to the beam. The metal sheet defined a finite length of gas radiator, and reflected the Cerenkov cone of radiation toward the accelerator but rotated by an angle 2<t> from the beam line. A microwave X-band antenna and crystal detector with response from 7 to above 12 GHz could be moved across the (reflected) Cerenkov cone as a probe. As mentioned earlier, the series A measurements showed the radiation is produced at the bunch repetition rate and its harmonics. Series B measurements performed with several antennas always indicated a broadened Cerenkov cone with strong radiation occuring at angles up to 10°, well beyond the predicted Cerenkov angle of 1.3°. 18 Since a broad band detector was used it was impossible to verify the prediction (see eq . 38) that the broadening cf the cone should depend on the harmonic number. However, it should be noted that incoherent radiation by a beam of lu A at 9 c = 1.3° for a 1 meter path in air would be about 10~^ watts at microwave frequencies so that observation of any signal by either method A or B clearly demonstrated coherent radiation by the electron bunches . Many of the concepts were clearly notea by Jelly in his treatise (Jelly^, Section 3.4 especially). The form factor was noted but a general expression was not given. In fact, the form factor quoted by Jelly represents the special case of a uniform line charge of length L' with a projected length L=L'cos9 c in the direction of the radiation. The coherence of the radiation from the bunch was noted but no broadening of the: cone nor harmonic structure were developed. Casey, Yen and Kaprielian-^ considered an apparently related problem in Cerenkov radiation, in whj ch a single electron passes through a dielectric medium, where a spatially periodic term is added to the dielectric constant. The result is radiation occurring even for electrons which do not exceed the velocity of light in the medium, and at angles other than the Cerenkov cone angle. The non-Cerenkov part of the radiation is attributed to transition radiation. In the present paper, the transition radiation associated with the gas cell boundaries is included, and radiation appears outside the Cerenkov cone. 19 If the electron velocity were lower so that v/c were close to but less than unity, the peak in I would be pushed to the left in Fig. 1, such that cos 9 c = v/c would be larger that 1. But. the tails of the diffraction function I would extend into the physical range 1 _< cos 8 <_ - 1 , and this would be called transition radiation and he ascribed to the passage of the electrons through the boundaries of the gas cell. Now return to the case v/c > i, with the situation as shown in Fig. 1. The radiation is then a combination of Cerenkov and transition radiation. The formalism of Reference 12 does admit a decomposition into the two types of radiation, but is inherently much more cumbersome. As a final remark, one might extend the analysis further in the region near w ^ . Consider electron bunches emitted frcm a travelling wave Linac, which could be 1 cm long spaced 10 cm apart. Let these bunches enter the wiggler magnet of a free electron laser (FEL) . Then, if gain occurs, the 1 cm bunches would be subdivided into bunches of a finer scale, with the spatial r-cale appropriate to the output wavelength of the FEL. 13 If the (partially) bunched beam from the FEL were passed into a gas Cerenkov cell, then the observed radiation should be reinforced because cf partial coherence, at the FEL bunch frequency and harmonics. This would lead to bumps in the spectrum in the region near oj ^ . 20 ACKNOWLEDGEMENTS This work was made possible by the Office of Naval Research W. B. Zeleny provided much insight into the theory for point charges, and the experiments performed by Ahmet Saglern, Lt . , Turkish Navy, provide preliminary substantiation of the work. 21 REFERENCES 1. Cerenkov, P., Dcklady Akad-Nauk, S.S.S.R., 2 451, (1934). 2. Frank, I.M. and Tamm, I., Doklady Akad-Nauk 3.S.S.R., L4 109, (1937) . 3. Jelly, J. V., "Cerenkov Radiation and its Applications". Pergammon Press, London, (1958). 4. Chamberlain, 0., E. Segre, C. Wiegand and T. Ypsilantis, Phys . Rev. , 100 947, (1955) . 5. E'anos, M.J., J. Appl. Phys, 26_ 1 , (19 55). 6. Laskinsky H. "Cerenkov Radiation at Microwave Frequencies" in advances in Electronics and Electron Physics, L. Marton, (ed) Academic Press, New York, Vol. XIV, pp. 265, (1961). 7. Piestrup, M.A., R.A. Powell, G.B. Rothbart, C.K. Chen and R.H. Fantell, Appl. Phys. Lett., 28. 92, (1976). S. Vialsh, J.S., T.C. Marshall and 3. P. Schlesinger, Phvs . Fluids, 20 709 1977. 9. Valsh, J.E. "Cerenkov and Cerenkov-Raman Radiation Sources" in P hysics of Quantum Electronics , S. Jacobs, M. Sargent and M. Scully (ed) Vol 7, Addison Wesley, r eading, Mass., (1979). 10. Jackson, J.D., "Classical Electrodynamics", (2 nc ^ Edition) John Wiley, New York 1975 (pp.4) Sec. 9.2 pp. 394ff. 11. Hofstadter , R. , Ann. Rev. Nuc Sci, Vol 7. pp. 2 31, (1937). 12. Casey, K.F., C. Yeh and Z.A. Keprielian, Phvs. Rev. 140 B 768 (1965) . 13. Hopf, F.A., P. Meystre and M.O. Scully, Phys. Rev. Lett 3_7 1215 (1976) . 22 r=TT/2Z J! *G * e Tf ^i igure 1. Qualitative Behavior of the Function I 2 (Z'). Both the function G, from Eq. 25 in the text, and the emission angle are displayed as independent variables. G' and G" are upper and lower limits. 23 Pu » OJ Fiqure 2. Schematic Behavior of Power Emitted as a function of Angular Frequency 24 F 3 Eo o - — r ♦2 AY W o i — i — 2TT -5- ^; Figure 3. Structure of charge pulse from a travelling wave accelerator. y is the phase angle of an electron relative to the peak of the travelling wave accelerating field. Electrons in the range +A^are passed by magnetic deflection system. 25 APPENDIX A DERIVATION OF CERENKOV RADIATION FOR A SINGLE PULSE OF CHARGE . Let the pulse be described by p' (r,t) = p ' Q (r- vt) Al Both k z and go are continuous variables in this case; "v is again along the z axis. If we expand in terras of a four dimensional fourier integral, p'(r,t) = l/(2ir) 4 e 1 ^ " k - r ) o ' (k, w) d 3 kdoj A2 It may be shown that the condition Ai gives: p'(k,aj) = 2tt6(o) - k z v) p' Q (k) A3 where p' (k) is the three dimensional spatial tranform of p 1 evaluated at t = o. All the fields have fourier integral rather than fouries series expansions and the energy radiated per unit solid angle become 2 ^ ~ £ X ] - -, u f 2, dt n • S = t" ,,,, 2 — (o dw 2tt (4tt) c J _ 'Jd 3 r' J df e i(ct'-n.?')o)/c. x ?( *, tl) | A4 m |2 W ("> , n ) do) Al The integrand is a symmetric function of co so that W <-'*-^3*"\ffl) r * w icu(t' -n-r' /c) ^ £,«»■, . , , e nxJ(r ! ,t') A5 where 1 y ,* -, 2.,2 = — -, — co vn x v) M , - 3 c lOTT M = ,3 , , Mcot'-n-r /c) ,,-"", ,, \ d r d-c e p (r , t ) A6 Now we may write p'(r',t') in a fourier integral representation •(r\t') (2ir) J. *3. , , , ,,.-*, M -i(o) 't'-k' -r ') a k dco p ( .< , co ) e A7 Inserting eq . A3 into eq. A7 and the result into eq A6, the integral over d^ ' involves only exponentials and yields (2rr)3 o3(y v ' _ om/c) , so that eq A6 becomes II d 3 k'dco'e i(a) " a) " )t, 6 3 (k' - ncu/c: 5 (aj'-k. 'v)p' (k') Now the integral over go' may be done; because of the 6 function, to' is evaluated at k z 'v. M = /dt^AV i i L(co-k v) t £.3.^, a. . .,,-*.. z (S (k ' - con/c) p ' (k ' ) -v o A2 Now do the integrals over k x ' , k y ' and k z ', noting that k ' appears in the exponential, but k ' and k ' do not, u _ [,, icot' -iwt'n M = / dt e e : v / c ' / / / / s ; p (am /c,am /C/um /c) ~ x y z This may be written as M = dt e p (oon/c) A8 where H = i-n v/c z A9 If we let the time interval be finite, from -T to + T, the integral is easily done: 2_ iM = — sin ojHTPo (nco/c) A10 M = 4T sin cjHt I Po (noo/c) | A ]_l (coHT) 2 A3 This result, eq. All may be inserted into A5 for oo • The factor n x v is just sin 9 where 9 is the angle between the radiation and the beam axis. W(oj, n ) = —- — -H- (a sin Z 9 4T Z sin coHT' l p ' (nca/c) | Al3 16 * C (wHT) 2 ' W is the energy radiated per unit solid angle per unit angular frequency, co • To proceed to the total energy, multiply by d ft (solid angle) and integrate. But n z = cos 9 so that dQ may be related to cH : d ft = d(cos 8)dft= - — dHdft A , ,, v A14 Tii'i functions in eq . A13 do not contain <j> so that integration over G yields 2 it. Thus: wi , j n 1 y 2 m 2f . 2 , ',2 sin coHT ,„ A(co,nidft = — co T /sin 9|p | dH 5 2rr 2 ~ (ooHT) 2 The sm 2 goHT / (coHT) 2 factor in the integral is peaked at H = 0, c which by eq. A9 is at n z = cos 9 = — , or the usual Cerenkov angle, 9 c . This function is more strongly peaked about H = for large values of T, and in fact, for large T we may evaluate sin^ and P ' at the point corresponding to H =0. Then the integral oo J 2 2 dx sin (ax) /(ax) = rr/a A4 may be used to evaluate eq . A15, yielding ffwdfi = i- -| 2Tsin 2 9c|p;(Vc)| 2 JJ A16 The emmission was assumed to occur in a time interval from -T to •KT ; accordingly dividing by 2T yields a rate of emmission, and multiplying by v converts to emmission per unit path length. Thus we obtain, for the large T limit: dxdco GO) = — ujdco sin 8 |o'(nco/c)| A17 where d^E/^^ is the energy emitted per unit path length per unit angular frequency range co . The corresponding expression for T not. large is H" d 2 E , U , /coT -= — -=— dco = -: — go dco — dxdco 4tt it sin 9jp * (noj/c) J sin^coHT H' (coHT) Aia where H" and H' are the value of H corresponding to 9=0 and 5 = tt respectively. Equations Al 7 and A18 then describe the energy radiated per unit path length and per unit angular frequency range. For the non periodic (single) pulse the radiation has a continuous freauency spectrum. For a point charae q, p' (k) is identically q and the usual Cerenkov formula is obtained. Equation A17 is quoted by Jelly, but only with the form factor corresponding to a uniform line charge of length L' . APPENDIX B DERIVATION OF EQUATION 7 Equation 7 is derived for the case in which J(r,t) is — » expanded in fourier series. Let the fourier coefficient for A be given by: T r A(r\o)) = i J dt A(r\t) e icut B1 o Assume that the green's function solution for A(r,t) is given: (r,t) = uj r /Td 3 r' | dt' J(r',t) D(r - r', t - t) 32 where D(r,t) - ~ Sit - r/c) Let the current density be expanded in a fourier series B3 J(?\f ) =Z e- iw ' t, J(?',a J ) B4 co' Then insert 32, B3 and B4 into 31 to obtain t (*.«> = H. J dt e 1 ^ JjfdV Jc.f ^ ,^— p- B5 e ia),t ' J(r« ,03') _> z Bl Do -ioi 't' Jl dt ' , note that t' appears in the <5 function and in The result is t' is evaluated at t' = t - j r-r 1 l/c. T u A(r,co) = — ~ T dt e loot d J r' 1 1 4tt i-» •»' r-r B6 I 0) -ico 't ico ' j r-r ' I /c ^, -^, 1, e e ' ' J(r ,a> ) Do the integral on t, note that T kj dt •" L (co-co ') L = 6 CO CO B7 Then do the sum on co ' A(r ,co; = 4tt d r J(r ,co; e ico I r-r 1 /c r-r 38 This proves the desired result, B8 is equation 7 as used in the main text. Appendix C TEMPORAL STRUCTURE OF THE ELECTRON PULSE FROM A TRAVELLING WAVE ACCELERATOR. Assume that the energy of a single electron emerging from a linac with phase $ relative to the travelling wave field is CI E = E cos-J; o This relation is shown on fig. 3, along with some dots representing electrons rear the maximum energy E c , with phases clustered about 'b = o and ty - 2^. Two bunches, separated by a phase difference of 2t, are separated by a time T-i = 1/f where f is the accelerator frequency, which is f Q = 2.85 x 10^ Hz for a typical S-band accelerator of the Stanford type. If a deflection system with energy resolution slit passes only energies E from E Q to E Q - E the corresponding range of phase Aip is C2 AE = E - E = E (1- cosAijj) o c For AiJj small, this reduces to AE = (Ai|;) 2 c3 E o " 2 CI The temporal pulse length T2 is T n = 2A<|; T,/2 C4 2 x or T_ = T • 2u^/2tt If C3 is used to evaluate Aii> in terms of the fractional energy resolution AE/E Q O 7T CD For 1% energy resolution, 1 , 2 /T 1 is about 1/20. The electrons thus emerge in short bunches, and the charge and current, when expressed in a fourier expansion, should have very strong harmonic content up to and aoove the 20th harmonic. C2. Appendix D FORM FACTORS This section provides details and examples of form factors for various charge distributions. From the main text, F differs only from P, the fourier transform of P, by the total charge q of the bunch, so that for k = o, F reduces to unity. Thus we define £ /// d3r P(r)e i ^'" F(k) = £ / M d J r Pirje^ 1 L q For spherically symmetric charge distributions , let k*r = kru, where u is the cosine of the angle between k and r. In spherical coordinates, d^r = dudyr 2 dr. Then we find, CO F ( k ) =1±I / dr r p ( r ) r~< 1, -J sin Kr D2 q k For k very small, sin x may be replaced by x - x 3 /6 and we have *° F(k) = 1 ±1 / dr r p(r)[kr - k 3 r 3 /6] D 3 q k J Then the two terms in the square bracket lead to separate integrals, the first term being unity and the second is similar to the integral used to calculate the mean square radius, <r~>, exceot for a factor k 2 /6. Thus we have F(k) = 1 - k 2 <r 2 >/6 D\ D4 For a uniform spherical charge distribution of radius R, as well a a spherical shell of radius k, the integral D2 may be performed easily F(k) = 3 (sin kR - kR cos kR) (Solid sphere) (kR) T D5 F(k) = I sin(kR) (Spherical shell) kR D6 For a line charge concentrated on the z axis, we may return to Dl and let p(r) = 5(x) <5(y) p"(z), so that F(k) = 1 dz p"(z) e ikz (line charge) D7 Q F(k) - j sin ( _) (uniform line charge kZ 2 of length Z) D8 Distorted spherical symmetry may be said to occur if the scale transformation z' = pz serves to make p spherically symmetric in tine prime system. Let F g be the form factor calculated by 2 in the prime frame. It is simple to show that F(k x , k y , k z ) = F s (k^, k^, k^/p) D9 DZ DISTRIBUTION LIST Office of Naval Research CAPT. M.A. Howard, Code 210 800 N. Quincy Arlington, VA. 22217 Office of Naval Research CDR James Offutt 1030 E. Green St . Pasadena, CA. 91106 Library, Naval Postgraduate School Office of Research Administration NPS, Code 012A Monterey, CA. 93940 F.R. Buskirk & J.R. Neighbours 20 Naval Postgraduate School Physics Dept . , Code 61 Monterey, CA. 93940 DUDLEY KNOX LIBRARY- RESEARCH , REPORTS 5 6853 01057876 8 U21041